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2006 Fiscal Year Final Research Report Summary

Research of fluctuations of phase boundaries in large interacting systems from the probabilistic point of view

Research Project

Project/Area Number 15340032
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKobe University

Principal Investigator

HIGUCHI Yasunari  Kobe University, Faculty of Science, Professor, 理学部, 教授 (60112075)

Co-Investigator(Kenkyū-buntansha) FUKUYAMA Katusi  Kobe University, Faculty of Science, Professor, 理学部, 教授 (60218956)
ADACHI Tadayoshi  Kobe University, Faculty of Science, Associate Professor, 理学部, 助教授 (30281158)
WATANABE Kiyoshi  Kobe University, Faculty of Science, Associate Professor, 理学部, 助教授 (60091245)
YOSHIDA Nobuo  Kyoto University, Garduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (40240303)
MURAI Joshin  Okayama University, Graduatte School of Humanities and Social Sciences, Assistant Professor, 大学院社会文化科学研究科, 助手 (00294447)
Project Period (FY) 2003 – 2006
Keywordscentral limit theorem / phase boundary / conditional limit theorem / Widom-Rowlinson model / percolation / Sierpinski carpet lattice / sharp transition
Research Abstract

The main aim of this research is to understand the fluctuation of phase boundaries appearing in many mathematical models of phase transitions from the probabilistic view point. Essentially, we could do it only for the Widom-Rowlinson model in two dimensions as a conditional central limit theorem for phase boundaries. However, this type of phenomenon is now well understood during these four years by the works of Ioffe, Bodineau and others. There still remains to be understood related to this problem but we understand that the main problem is solved.
Our second aim was to understand the transition mechanism in percolation when the underlying graph has no translations which act as group of automorphisms of the underlying graph. A typical problem is in the case where the graph has infinitely ramified fractal structure. As an example, percolation in the Sierpinski carpet lattice has not been understood well. We could prove that percolation is sharp for this model. This has been open since 1997. The sharpness of the percolation transition is understood as
1.there is only one critical point
2.below the critical point, the connectivity function decays exponentially
3. above the critical point, the infinite cluster is unique and the dual connectivity decays exponentially with respect to the dual graph distance.

  • Research Products

    (26 results)

All 2007 2006 2005 2004 2003 Other

All Journal Article (24 results) Book (2 results)

  • [Journal Article] Analyticity of survival probability for contact process around t=02007

    • Author(s)
      V.Belitsky, Y.Higuchi, N.Konno, N.Sugimine
    • Journal Title

      Analysis and Applications 5

      Pages: 67-76

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Analyticity of survival probability for contact process around t=02007

    • Author(s)
      V.Belitsky, Y.Higuchi, N.Konno, N.Sugimine
    • Journal Title

      Analysis and Applications vol.5, No.1

      Pages: 67-76

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Gap series and functions of bounded variation2006

    • Author(s)
      K.Fukuyama
    • Journal Title

      Acta Mathematica Hungarica 110

      Pages: 175-191

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Directed polymers in random environment are diffusive at weak disorder2006

    • Author(s)
      F.Comets, N.Yoshida
    • Journal Title

      Annals of Probability 34

      Pages: 1746-1770

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Gap series and functions of bounded variation2006

    • Author(s)
      K.Fukuyama
    • Journal Title

      Acta Mathematica Hungarica Vol. 110

      Pages: 175-191

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Directed polymers in random environment are diffusive at weak disorder2006

    • Author(s)
      F.Comets, N.Yoshida
    • Journal Title

      Annals of Probability Vol. 34

      Pages: 1746-1770

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Brownian directed polymers in random environment,2005

    • Author(s)
      F.Comets, N.Yoshida
    • Journal Title

      Communications in Mathematical Physics Vol. 254

      Pages: 257-287

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Some results on the phase structure of the two-dimensional Widom-Rowlinson model2004

    • Author(s)
      Y.Higuchi, M.Takei
    • Journal Title

      Osaka Journal of Mathematics 41

      Pages: 237-255

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] The Dobrushin-Hryniv theory for the two-dimensional latice Widom-Rowlinson model2004

    • Author(s)
      Y.Higuchi, J.Murai, J.Wang
    • Journal Title

      Advanced Studies in Pure Mathematics 39

      Pages: 233-281

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Some results on the phase structure of the two-dimensional Widom-Rowlinson model2004

    • Author(s)
      Y.Higuchi, M.Takei
    • Journal Title

      Osaka Journal of Mathematics Vol. 41

      Pages: 237-255

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] An asymptotic property of gap series III2004

    • Author(s)
      K.Fukuyama
    • Journal Title

      Acta Mathematica Hungarica 102

      Pages: 97-106

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A concrete upper bound in the uniform law of the iterated logarithm2004

    • Author(s)
      K.Fukuyama
    • Journal Title

      Studia Scientiarum Mathematicarum Hungarica 41

      Pages: 339-346

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Infinite systems of non-colliding Brownian particles2004

    • Author(s)
      M.katori, H.Tanemura
    • Journal Title

      Advanced Studies in Pure Mathematics Vol.39

      Pages: 283-306

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Symmetry of matrix-valued stochastic processes and non-colliding diffusion particle systems2004

    • Author(s)
      M.Katori, T.Tanemura
    • Journal Title

      Journal of Mathematical Physics Vol. 45

      Pages: 3058-3085

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] An asymptotic property of gap series II2003

    • Author(s)
      K.Fukuyama, B.Petit
    • Journal Title

      Acta Mathematica Hungarica Vol. 98

      Pages: 245-258

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Localization transition of d-friendly walkers2003

    • Author(s)
      H.Tanemura, N.Yoshida
    • Journal Title

      Probability Theory and Related Fields Vol. 125

      Pages: 593-608

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Noncolliding Brownian motions and Harish-Chandra formula2003

    • Author(s)
      M.Katori, H.Tanemura
    • Journal Title

      Electronic Communications in Probability Vol. 8

      Pages: 112-121

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Functional central limit theorems for vicious walkers2003

    • Author(s)
      M.Katori, H.Tanemura
    • Journal Title

      Stochastics and Stochastics Reports Vol. 75

      Pages: 369-390

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Directed polymers in random environment : path localization and strong disorder2003

    • Author(s)
      F.Commets, T.Shiga, N.Yoshida
    • Journal Title

      Bernoulli Vol.9, (3)

      Pages: 705-723

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] The Dobrushin-Hryniv theory for the two-dimensional lattice Widom-Rowlinson model

    • Author(s)
      Y.Higuchi, J.Murai, J.Wang
    • Journal Title

      Advanced Studies in Pure Mathematics Vol. 39

      Pages: 233-281

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On the Mourre estimates for three body Schroedinger operators in a constant magnetic field

    • Author(s)
      T.Adachi
    • Journal Title

      Funccialaj Ekvacioj (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Asymptotic completeness for N-body quantum systems with long-range interactions in a time-periodic electric field

    • Author(s)
      T.Adachi
    • Journal Title

      Communications in Mathematical Physics (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On multidimensional inverse scattering for Stark Hamiltonians

    • Author(s)
      T.Adachi, K.Maehara
    • Journal Title

      Journal of Mathematical Physics (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Infinite systems of non-colliding generalized meanders and Riemann-Liouville Differintegrals

    • Author(s)
      M.Katori, H.Tanemura
    • Journal Title

      Probability Theory and Related Fields (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] 確率論教程シリーズ3 確率過程入門2006

    • Author(s)
      西尾真喜子, 樋口保成
    • Total Pages
      239
    • Publisher
      培風館
    • Description
      「研究成果報告書概要(和文)」より
  • [Book] 確率論教程シリーズ6 統計力学2006

    • Author(s)
      黒田耕嗣, 樋口保成
    • Total Pages
      226
    • Publisher
      培風館
    • Description
      「研究成果報告書概要(和文)」より

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Published: 2008-05-27  

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